Where can I find generating functions for orthogonal polynomials in two variables?
Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for
$$
\sum_{n=0}^\infty \frac{n!}{\Gamma(n+\alpha+1)} L_n^\alpha(x) L_n^\alpha(y) t^n
$$
as an explicit function of $x,y,t$ (it involves a Bessel function). However, I need to sum over the upper index (when it is an integer), to find
$$
\sum_{k=0}^\infty \frac{n!}{(n+k)!} L_n^k(x) L_n^k(y) t^k = {??}
$$
in a closed form $F(x,y,t)$, for a fixed positive integer $n$. Are generating functions of this kind to be found in the literature? (Not on DLMF, as far as I can see.) A closed form for the previous sum would be great, a pointer to a suitable article would be better.

1 Answer
1

I think that http://arxiv.org/pdf/math-ph/0409066v1 (Multivariate Orthogonal Polynomials (symbolically) page 15, has the representation you are looking for [whether it will help you compute your sum, I am not sure, but maybe their Maple package will do the thinking for you?

Not really, but its bibliography pointed to an article of Baker and Forrester (Commun. Math. Phys. 188, 1997) that has a nice general theory of multivariate generating functions. Sadly, nothing there about summation on the upper index.
–
jvarillyDec 9 '11 at 14:47